Explicit estimates for \(\delta _g\)  and \(\varepsilon _g\)  are given at the end of Section . Note that the result of Otal and Rosas cannot be extended beyond \(1/4\) , as a hyperbolic surface of genus \(g\)  can have as many eigenvalues as we want in the interval \((1/4,1/4+\mu )\)  for any \(\mu >0\)  if it has a sufficiently short closed geodesic (see the proof of Theorem 8.1.2 in [1]}). However, if one assumes a lower bound \(c>0\)  on the systole of \(S\) , then the upper bound of \(2g-3\)  does extend to an interval of the form \((0,1/4+\mu _{g,c}]\)  for some explicit \(\mu _{g,c}>0\)  [2]}. A related improvement says that if \(S\)  is a closed hyperbolic surface of genus \(g\)  with systole larger than \(3.46\) , then \(S\)  has at most \(2g-4\)  Laplacian eigenvalues in \((0,1/4]\)  [3]}.
